Bivariate distributions in statistical spectroscopy studies: Fixed-J level densities,fixed-J averages and spin cut-off factors

Analytical studies with the TBRE ensemble, supplemented by numerical calculations with realistic interactions show that for large particle numbers, the bivariate cumulants (k _rs ) defined for the Hamiltonian operator (H) and theJ _z operator, are very small; ∥k _rs ∥?0.3 for 3≦r+s≦6. As a result the expansions around a bivariate normal density are meaningful for the fixed-M densities (ρ(E,M)). We adopt a bivariate Edgeworth expansion forρ(E,M) and give a compact form for the same. Finally using thisρ(E,M) (which also define fixed-J densities uniquely), new series expansions are given for fixed-M (and hence for fixed-J) averages of the powers ofH and also for the spin cut-off factors.